‎let $g=(v(g),e(g))$ be a simple connected graph with vertex set $v(g)$ and edge‎ ‎set $e(g)$‎. ‎the (first) edge-hyper wiener index of the graph $g$ is defined as‎: ‎$$ww_{e}(g)=sum_{{f,g}subseteq e(g)}(d_{e}(f,g|g)+d_{e}^{2}(f,g|g))=frac{1}{2}sum_{fin e(g)}(d_{e}(f|g)+d^{2}_{e}(f|g)),$$‎ ‎where $d_{e}(f,g|g)$ denotes the distance between the edges $f=xy$ and $g=uv$ in $e(g)$ and $d_{e}(f|g)=s...

In this research paper, we present a novel frame work for handling $m$-polar information by combining the theory of $m-$polar fuzzy  sets with graphs. We introduce certain types of edge regular $m-$polar fuzzy graphs and edge irregular $m-$polar fuzzy graphs. We describe some useful properties of edge regular, strongly edge irregular and strongly edge totally irregular $m-$polar fuzzy graphs. W...

The edge tenacity Te(G) of a graph G is dened as:Te(G) = min {[|X|+τ(G-X)]/[ω(G-X)-1]|X ⊆ E(G) and ω(G-X) > 1} where the minimum is taken over every edge-cutset X that separates G into ω(G - X) components, and by τ(G - X) we denote the order of a largest component of G. The objective of this paper is to determine this quantity for split graphs. Let G = (Z; I; E) be a noncomplete connected split...

The numbers of 2-connected and 2-edge-connected Steinhaus graphs with n vertices are determined for n¿ 6. The possible generating strings of 2-connected and 2-edge-connected Steinhaus graphs are classi0ed. c © 2002 Elsevier Science B.V. All rights reserved.

Let G 1⁄4 ðV ;EÞ be a k-regular graph with connectivity j and edge connectivity k. G is maximum connected if j 1⁄4 k, and G is maximum edge connected if k 1⁄4 k. Moreover, G is super-connected if it is a complete graph, or it is maximum connected and every minimum vertex cut is fxjðv; xÞ 2 Eg for some vertex v 2 V ; and G is super-edge-connected if it is maximum edge connected and every minimum...

We consider the problems of finding the minimum-weight 2-connected spanning subgraph in edge-weighted planar graphs and its variations. We first give a PTAS for the problem of finding minimum-weight 2-edge-connected spanning subgraphs where duplicate edges are allowed. Then we present a new greedy spanner construction for edge-weighted planar graphs. From this we derive quasi-polynomial time ap...

For a graph G, the P2-path graph, P2(G), has for vertices the set of all paths of length 2 in G. Two vertices are connected when their union is a path or a cycle of length 3. We present lower bounds on the edge-connectivity, (P2(G)) of a connected graph G and give conditions for maximum connectivity. A maximally edge-connected graph is superif each minimum edge cut is trivial, and it is optimum...